Abstract
We develop efficient methods for deterministic computations with semi-algebraic sets and apply them to the problem of counting points on curves and Abelian varieties over finite fields. For Abelian varieties of dimension g in projective N space overFq, we improve Pila’s result and show that the problem can be solved in O((logq)δ) time where δ is polynomial in g as well as in N. For hyperelliptic curves of genus g overFq we show that the number of rational points on the curve and the number of rational points on its Jacobian can be computed in (logq)O(g2logg)time.
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